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Unformatted text preview: 8.1 Underlying Graphs of Various Classes of Digraphs 423 B i A i C i C j C p A p B p A j B j Figure 8.3 Implication classes for orientations of a graph G as a local tournament digraph.The sets C i ,C j ,C p denote distinct connected components of G . For each component a bipartition A r ,B r is shown. The edges shown inside C i form one implication class and the edges shown between C j and C p form another implication class. Theorem 8.1.13 (Huang)  Let G be a proper circular-arc graph which is reduced (that is, every edge is unbalanced), let ¯ G denote the complement graph of G and let C 1 ,...,C k denote the connected components of ¯ G . (a) If ¯ G is not bipartite, then k = 1 and (up to a full reversal) G has only one orientation as a locally tournament digraph, namely the round ori- entation. (b) If ¯ G is bipartite then every orientation of G as a locally tournament digraph can be obtained from the round locally tournament digraph ori- entation D of G by repeatedly applying one of the following operations: (I) reverse all arcs in D that go between two different C i ’s, (II) reverse all arcs in D that have both ends inside some C i . ut It is also possible to derive a similar result characterizing all possible orientations of G as a locally semicomplete digraph. We refer the reader to  for the details. As an example of the power of Huang’s result (Theorems 8.1.12 and 8.1.13) we state and prove the following corollary which was implicitly stated in  (see also Exercise 4.33). Corollary 8.1.14 If D is a locally tournament digraph such that UG ( D ) is not bipartite, then D = R [ S 1 ,...,S r ] , where R is a round locally tournament digraph on r vertices and each S i is a strong tournament. Proof: If UG ( D ) is reduced, then this follows immediately from Theorem 8.1.13, because according to Theorem 8.1.13, there is only one possible lo- cally tournament digraph orientation of UG ( D ). So suppose that UG ( D ) is not reduced. By Lemma 8.1.11, UG ( D ) = H [ K a 1 ,...,K a h ], h = | V ( H ) | , where H is a reduced proper circular arc graph, each K a i is a complete graph, 424 8. Orientations of Graphs and some a i ≥ 2. Because we can obtain an isomorphic copy of H as a sub- graph of UG ( D ) by choosing an arbitrary vertex from each K a i , we conclude, from Theorem 8.1.12, that in D all arcs between two distinct K a i , K a j have the same direction (note that H is non-bipartite). Thus D = R [ S 1 ,...,S r ], where (up to reversal of all arcs) R is the unique round locally tournament digraph orientation of H and each S i is the tournament D h V ( K a i ) i . Note that D h V ( K a i ) i may not be a strong tournament, but according to Corol- lary 4.11.7 we can find a round decomposition of D so that this is the case....
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This document was uploaded on 08/10/2011.
- Spring '11